I need to prove that for a given function $ f $ continuous on $ [a,\infty)$ with $ a \in \mathbb{R}$, $ \exists\lim_{x \to \infty} f(x) \implies f $ is unif. continuous on $[a, \infty) $, which isn't to complicated after playing with epsilons and deltas. Now, I have to determine whether the reciprocal is true or not. That means determining if the following statement is true or not:
$ f $ unif. cont. $\implies \exists\lim_{x \to \infty} f(x)$
I have simply come up with a counterexample leveraging the fact that $ f(x) = x$ is unif. cont. but the limit doesn't exist. My question is: How do I prove the reciprocal is not true rigorously with epsilons and deltas? I have attempted it, with little success.
All comments and critics are welcome
Given a continuous function $f$, by the Weierstrass theorem we know that for every $a<b\in \mathbb{R}, f$ is uniformly continuous $[a,b]$. if.
the function $f(x)=x$ is not uniformly continuous in $[a,\infty)$, exactly because $lim_{x\to \infty}f(x) \to \infty$.
This means that if the limit of a function $f$ as $x$ approaches infinity exists and and is finite, then $f$ is a unif cont function in $[a,\infty)$.